GÃ¶del's Incompleteness Theorem

At the beginning of the 20th century, mathematicians were beginning to feel that it was going to be possible to unify the whole of mathematics, and to catalogue the complete subject. An attempt was made by Bertrand Russell (1872 - 1970) and , Alfred North Whitehead (1876 - 1947) in their Principia Mathematica, where they attempted to reduce the whole of known mathematics to set theory. Slightly later, , David Hilbert (1862 - 1943) outlined the Hilbert Programme for mathematics, in which an attempt was to be made to classify the whole of mathematics and to give a method by which the truth or falsity of theorems written in symbolic logic could be determined in a purely mechanistic way. This programme was ambitiousâ€”it would mean the end of the task of mathematics if it were to be completed. (The idea that it might be possible was suggested by the similarity of the proofs of theorems from many disparate parts of mathematics.)

However, the Hilbert programme was hardly started when it was dealt a deadly blow. A young mathematician, , Kurt GÃ¶del (1906 - 1978) proved his incompleteness theorem, which showed that the programme was impossible to complete. GÃ¶del was studying the logic of Peano arithmetic, an axiomatization of the arithmetic of the natural numbers. He realized that every sentence of symbolic logic could be encoded with a unique number, and that it was therefore possible to write down sentences in symbolic logic that would mean things like â€˜2 + 2 = 4 is provableâ€™. He then looked at sentences of the form â€˜The sentence with number n is trueâ€™ and found that, using the method of recursion, it was possible to find a sentence with number n which, when translated, essentially said, â€˜the sentence with number n is true but not provableâ€™, in other words, â€˜this sentence is true but not provableâ€™. So he had shown that there were some true statements which it was impossible to prove, in Peano arithmetic. The same is true of any reasonably powerful mathematical system, and so the Hilbert programme is completely unattainable. SMcL